Then \(H^\sharp (k_x)\)  is symmetric (i.e. formally self-adjoint) on the initial dense domain \(C_c^\infty ((0,\infty );\mathbb {C}^3)\subset L^2((0,\infty );\mathbb {C}^3)\) , and we use the same symbol for its closure. It may be extended to a self-adjoint operator by specifying appropriate boundary conditions at \(y=0\) . As in [1]}, we shall focus on the following family of boundary conditions:
\(v|_{y=0} = 0, \qquad ({\mathrm {i}}k_x u + a \partial _y v)|_{y=0} = 0, \) 
